High-efficiency electro-optic modulators are widely used in optical communication systems to convert analog or digital signals from the electrical domain to the optical domain. The modulation efficiency is generally dictated by the Vπ, which is defined as the input voltage needed for the modulated arm or arms of the interferometer to accumulate a change in optical phase difference of π radians. The lower the Vπ, the better the modulation efficiency. Since Vπ, can generally be reduced by increasing the electrical-optical field interaction length, L, designers often use the length-independent product of Vπ and the modulation length, Vπ·L, as a figure of merit for comparative evaluation of modulators.
Many conventional optical modulators are lithium niobate Mach-Zehnder Interferometric modulator (MZI modulators). The arms of typical interferometers are often made from Titanium-diffused lithium niobate waveguides. The Vπ·L product can be calculated. For example, the Vπ·L product at λ=1.55 μm for a lithium niobate MZI modulator in which only one arm is driven by the modulation voltage can be calculated using the following expression:
                                          V            π                    ⁢          L                =                                            λ                                                n                  3                                ⁢                                  r                  33                                                      ⁢                          d              Γ                                =                                                                      1.55                  ⁢                                                                          ⁢                  µm                                                                      2.14                    3                                    ×                  30.8                  ×                                      10                                          -                      6                                                        ⁢                                                                          ⁢                                      µm                    /                    V                                                              ⁢                              δ                Γ                                      =                          0.51              ⁢                              d                Γ                            ⁢                              (                                  V                  ·                  cm                                )                                                                        (        1        )            where n is the typical lithium niobate refractive index, which is about 2.14, r33 is the magnitude of the relevant electro-optic (EO) tensor, which is about 30.8×10−6 μm/V, d is the gap between the electrodes in microns, and Γ is the electrical-optical field overlap integral. The electo-optical field overlap integral is defined by the following expression:
                    Γ        =                              d            V                    ⁢                                    ∫                                                ∫                  s                                ⁢                                                      E                    opt                    2                                    ⁢                                      E                    x                                    ⁢                                      ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                      ∫                                                ∫                  s                                ⁢                                                      E                    opt                    2                                    ⁢                                      ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                                          (        2        )            where V is the applied voltage, Eopt is the optical field, and Ex is the electric field that is polarizing along the optical axis of the lithium niobate crystal. A typical value for the Vπ·L product is 16.4 V·cm for a lithium niobate MZI modulator with a gap between the electrodes, d, equal to 20 μm and an electrical-optical field overlap integral, Γ equal to 62%.